Contents of "Dynamics of Composite Materials and Structures"

Preface

 

Chapter 1   Basic Elements on Laminate and Sandwich Composite

                   Materials

- 1.1    Constituents and Architecture of Composite Materials

- 1.1.1 The Constituents of Composite Materials

- 1.1.2 Laminate Composite Materials     

- 1.1.3 Sandwich Composites

- 1.2    Elastic Behaviour of Composite Materials          

- 1.2.1 Unidirectional Composite Materials

- 1.2.2 Orthotropic Composite Materials

- 1.2.3 Stress-Strain Relations for Off-Axis Layers

- 1.2.4 Plane Stress State

- 1.2.4.1 Two-Dimensional Stress State      

- 1.2.4.2 Elasticity Equations for Plane Stress

- 1.2.4.3 Elasticity Equations in Material Directions

- 1.2.4.4 Off-Axis Reduced Stiffness Constants

- 1.3    Basics of Laminate Theory

- 1.3.1 Introduction

- 1.3.2 Displacement Field

- 1.3.3 Resultants and Moments

- 1.3.3.1 In-Plane Resultants 

- 1.3.3.2 Transverse Shear Resultants

- 1.3.3.3 Resultant Moments

- 1.3.4  Fundamental Equations for Plates in the Case of First-Order

            Theory

- 1.4    Classical Laminate Theory

- 1.4.1 Assumptions of the Classical Theory of Laminates

- 1.4.2 Strain Field

- 1.4.3 Stress Field

- 1.4.3.1 General Expression 

- 1.4.3.2 Stress Field in the Case of Classical Laminate Theory

- 1.4.4 Resultants and Moments

- 1.4.5 Constitutive Equation of Laminate

- 1.4.6 Governing Equations

- 1.4.7 Boundary Equations

- 1.4.7.1 Basics

- 1.4.7.2 Simply Supported Edge

- 1.4.7.3 Clamped Edge

- 1.4.7.4 Free Edge

- 1.4.8 Energy Formulation of the Classical Laminate Theory

- 1.4.8.1 Strain Energy

- 1.4.8.2 Kinetic Energy

- 1.5    Laminate Theory Including the Transverse Shear Effects

- 1.5.1 Constitutive Equation

- 1.5.2 Governing Equations

- 1.5.3 Boundary Conditions

- 1.5.4 Introduction of Transverse Shear Coefficients

- 1.6.   Theory of Sandwich Plates

- 1.6.1 Introduction

- 1.6.2 Assumptions of the Sandwich Theory

- 1.6.3 Displacement Field 

- 1.6.4 Strain Field

- 1.6.5 Stress Field

- 1.6.6 Constitutive Equation

- 1.6.7 Fundamental Equations

 

Chapter 2   Dynamics of Systems with One Degree of Freedom

- 2.1    Equation of Motion of a System with One Degree of Freedom

- 2.2    Undamped Vibrations

- 2.2.1 Equation of Motion 

- 2.2.2 Free Vibrations

- 2.2.3 Forced Vibrations. Steady State

- 2.2.3.1 Case of a harmonic disturbing force

- 2.2.3.2 Case of a harmonic displacement of the spring end

- 2.3    Vibrations with Viscous Damping

- 2.3.1 Introduction

- 2.2.2 Equation of Motion with Viscous Damping

- 2.3.3 Free Vibrations

- 2.3.3.1 Characteristic Equation

- 2.3.3.2 Case of Low Damping

- 2.3.3.3 Case of High Damping

- 2.3.3.4 Critical Damping

- 2.3.4 Vibrations in the case of Harmonic Disturbing Force

- 2.3.4.1 Time Domain

- 2.3.4.2  Frequency Domain

- 2.3.4.3  Effect of the Frequency of the Disturbing Force

- 2.3.4.4  Damping Modelling using Complex Stiffness

- 2.3.5 Vibrations in the case of Periodic Disturbing Force

- 2.3.6 Vibrations in the case of Arbitrary Disturbing Force

- 2.4    Equivalent Viscous Damping Capacity

- 2.4.1 Introduction

- 2.4.2 Energy Dissipated in the case of Viscous Damping

- 2.4.3 Loss Factor and Specific Damping

- 2.4.4 Structural Damping

 

Chapter 3   Beam Bending and Cylindrical Bending Vibrations of

                   Undamped Laminate and Sandwich Materials

- 3.1    Introduction

- 3.2    Equation of Motion of Symmetric Laminate Beams

- 3.3    Natural Modes

- 3.3.1 Mode Shapes

- 3.3.2 Properties of the Mode Shapes

- 3.4    Natural Modes of Beams with Different End Conditions

- 3.4.1 Simply Supported Beam

- 3.4.2 Clamped Beam

- 3.4.3 Beam Clamped at One End and Simply Supported at the Other

- 3.4.4 Beam Clamped at One End and Free at the Other        
- 3.4.5 Beam with Two Free Ends

- 3.5    Normal Mode Analysis

- 3.5.1 Motion Equation in Normal Coordinates

- 3.5.2 Response to Initial Conditions

- 3.5.2.1 General Formulation         

- 3.5.2.2 Beam with Simply Supported Ends

- 3.5.2.3 Beam with Other End Conditions

- 3.5.3 Forced Response

- 3.5.3.1 General Formulation         

- 3.5.3.2  Beam with Simply Supported Ends

- 3.6    Cylindrical Bending Vibrations

- 3.6.1 Introduction

- 3.6.2 Classical Laminate Theory

- 3.6.2.1 Equations 

- 3.6.2.2 Plate Simply Supported

- 3.6.2.3 Plates with Other End Conditions

- 3.6.3 Effect of Transverse Shear

- 3.6.4 Cylindrical Vibrations of Sandwich Plates         

 

Chapter 4   Flexural Vibrations of Rectangular Laminate Plates 

- 4.1    Free Vibrations of Rectangular Orthotropic Plates Simply

           Supported along the Edges

- 4.2    Vibrations of Orthotropic Plates with Various Conditions along the

            Edges

- 4.2.1 General Expressions

- 4.2.2 Rayleigh’s Approximation

- 4.2.3Two-Term Approximation

- 4.2.4Orthotropic Plates with Simply Supported or Clamped Edges

- 4.3    Vibrations of Symmetric Laminate Plates

- 4.3.1 General Expressions

- 4.3.2 Symmetric Plates with Clamped or Free Edges

- 4.4    Vibrations of Non-symmetric Laminate Plates

- 4.4.1 Plate Constituted of an Antisymmetric Cross-Ply Laminate

- 4.4.2 Plate Constituted of an Angle-Ply Laminate

- 4.5    Evaluation of the Laminate Bending Stiffnesses by Analysis of

           Plate Vibrations     

- 4.5.1 Introduction

- 4.5.2 Experimental Features

- 4.5.2.1 Materials

- 4.5.2.2 Boundary Conditions

- 4.5.2.3 Experiment

- 4.5.3 Introduction to the Experimental Modal Analysis of

           Orthotropic Plates

- 4.5.3.1 Evaluation of the Natural Frequencies

- 4.5.3.2 Different Results

- 4.5.4 Experimental Results and Discussion in the case of Orthotropic

            Plates 



- 4.5.4.1Values of the Natural Frequencies

- 4.5.4.2 Determination of the Stiffnesses by an Iterative

- 4.5.4.3 Evaluation of the Stiffnesses from Rayleigh’s Approximation

- 4.5.4.4 Discussion of the Results and Conclusions

- 4.5.5 Case of Symmetric Laminates

- 4.5.5.1 Materials and Results

- 4.5.2.2 Evaluation of the Bending Stiffnesses from the Natural

              Frequencies and Mode Shapes

 

Chapter 5  Damping in Composite Materials

- 5.1    General Comments 

- 5.2    Damping in a Unidirectional Composite as Function of the

           Constituents

- 5.3.   Bending Vibrations of Damped Laminate Beams

- 5.3.1 Damping Modelling using Viscous Friction

- 5.3.2 Motion Equation in Normal Coordinates

- 5.3.3 Forced Harmonic Vibrations

- 5.3.4 Damping Modelling using Complex Stiffness

- 5.3.5 Beam Response to a Concentrated Loading

- 5.4    Evaluation of the Damping Properties of Orthotropic Beams as

           Function of Material Orientation

- 5.4.1 Energy Analysis of Beam Damping

- 5.4.1.1 Introduction

- 5.4.1.2 Adams-Bacon Approach

- 5.4.1.3 Ni-Adam Analysis

- 5.4.1.4 General Formulation of Damping

- 5.4.2 Complex Moduli

- 5.5    Evaluation of the Damping Properties of Plates as Function of

           Material Direction

- 5.5.1 Orthotropic Plates

- 5.5.1.1 Formulation

- 5.5.1.2 Procedure

- 5.5.2 Laminated Plates

- 5.5.3 Conclusion

 

Chapter 6  Experimental Investigation and Discussion on the Damping

                  Properties of Laminates 

- 6.1    Experimental Investigation in Literature

- 6.1.1 Experimental Processes for Evaluating Damping

- 6.1.2Experimental Results

- 6.2    Damping Analysis of Unidirectional Glass and Kevlar

           Fibre Composites

- 6.2.1 Introduction

- 6.2.2 Materials

- 6.2.3 Experimental Equipment

- 6.2.4 Analysis of the Experimental Results

- 6.2.4.1 Determination of the Constitutive Damping Parameters

- 6.2.4.2 Plate Damping Measurement

- 6.2.5 Choice of the Frequency Range for the Experimental

            Investigation

- 6.2.6 Experimental Results

- 6.2.6.1 Introduction

- 6.2.6.2 Stiffness

- 6.2.6.3 Damping

- 6.3 Comparison of Experimental Results and Models for

         Unidirectional Beam Damping

- 6.3.1 Models of Adams-Bacon and Ni-Addams

- 6.3.1.1 Introduction 

- 6.3.1.2 Glass Fibre Composites

- 6.3.1.3 Kevlar Fibre Composites

- 6.3.2 Complex Stiffness Model

- 6.3.3 Using the Ritz Method

- 6.3.3.1 Damping Parameters         

- 6.3.3.2 Influence of the Width of the Beams

- 6.3.3.3 Damping According to Modes of Beam Vibrations

- 6.4 Damping of Laminated Beams

- 6.5    Damping of Laminated Plates

- 6.5.1 Damping Investigation

- 6.5.2 Plates with One Edge Clamped and the Other Edges Free

- 6.5.3 Plates with Two Edges Clamped and the Other Edges Free

- 6.6    Longitudinal and Transverse Damping of Unidirectional Fibre

           Composites

- 6.6.1 Introduction

- 6.6.2 Longitudinal Damping

- 6.6.3 Transverse Damping

- 6.6.3.1 Formulation 

- 6.6.3.2 Application

- 6.7    Temperature Effect on the Damping Properties of Unidirectional

           Composites

- 6.7.1 Introduction

- 6.7.2 Materials and Experiment

- 6.7.3 Experimental Results

- 6.7.3.1 Matrix properties     

- 6.7.3.2 Composite Properties

- 6.7.3.3 Damping Evaluation Based on the Ritz Method

 

Chapter 7   Damping Analysis of Laminates with Interleaved



                  Viscoelastic Layers

- 7.1    Introduction

- 7.2    Damping Modelling of Orthotropic Laminates with Interleaved

           Viscoelastic Layers

- 7.2.1 Laminate Configurations

- 7.2.2 In-Plane Damping with Interleaved Viscoelastic Layers

- 7.2.2.1 Case of a Single Interlaminar Viscoelastic Layer

- 7.2.2.2 Case of Two Interlaminar Viscoelastic Layers

- 7.2.3 Considering the Transverse Shear Effects in the Case of a Single

            Interlaminar Viscoelastic Layer

- 7.2.3.1 Introduction

- 7.2.3.2 Transverse Shear Stresses in Layers

- 7.2.3.3 Strain Energy Stored in xz-Transverse Shear

- 7.2.3.4 Strain Energy Stored in yz-Transverse Shear

- 7.2.3.5 Laminate Damping with a Single Interleaved Viscoelastic Layer

              Including the Transverse Shear Effects

- 7.2.4 Considering the Transverse Shear Effects in the case of Two

            Interleaved Viscoelastic Layers

- 7.2.4.1 Case of Symmetric Laminate

- 7.2.4.2 Transverse Shear Stresses in the (x, z) plane

- 7.2.4.3 Transverse Shear Energies in the (x, z) plane

- 7.2.4.4 Transverse Shear Energies in the (y, z) plane

- 7.2.4.5  Laminate Damping with Two Interleaved Viscoelastic Layer

               Including the Transverse Shear Effects

- 7.2.5 Application to Angle-Ply Laminates

- 7.2.6 Laminates with External Viscoelastic Layers

- 7.2.7 Choice of the Basis Functions of the Ritz Method

- 7.3    Experimental Investigation and Discussion of Damping of

           Unidirectional Composites with Interleaved Viscoelastic Layers

- 7.3.1 Materials

- 7.3.2 Experimental Results

- 7.3.3 Analysis of the Experimental Results

- 7.3.3.1 Dynamic Properties of the Viscoelastic Layers

- 7.3.3.2 Damping of Unidirectional Glass Fibre Laminates with      

               Interleaved Viscoelastic Layers

- 7.4    Analysis of the Experimental Results Obtained in the case of Angle

           -Ply Laminates

- 7.5    Discussion

- 7.6    Conclusion

 

Chapter 8   Finite Element Method In the Dynamic Analysis of
                  Composite Structures

- 8.1    Principle of the Method

- 8.2    Formulation of Structural Elements

- 8.2.1 Isoparametric Finite Element Formulation

- 8.2.2 Example of a Four-Node Finite Element

- 8.2.2.1 Interpolation Functions

- 8.2.2.2 Strain Formulation

- 8.3    Laminate Element

- 8.3.1 Displacement Field

- 8.3.2 In-Plan Behaviour

- 8.3.3 Flexural Behaviour  

- 8.3.4 Transverse Shear Behaviour

- 8.3.5 Stress Formulation  

- 8.3.6 Energy Formulation

- 8.3.6.1 Strain Energy and Element Stiffness

- 8.3.6.2 Kinetic Energy

- 8.3.6.3 Work of the External Loads

- 8.4    Finite Element Dynamic Equation of Laminate Structure

 

Chapter 9   Solution of Dynamic Equation in Finite Element Analysis

- 9.1    Introduction

- 9.2    Direct Integration Methods

- 9.2.1 Principle

- 9.2.2 The Central Difference Method    

- 9.2.2.1 Formulation

- 9.2.2.2 Characteristics of the Central Difference Method

- 9.2.3 The Houbolt Method         

- 9.2.4 The Wilson θ Method

- 9.2.5 The Newmark Method

- 9.3    Mode Superposition

- 9.3.1 Introduction

- 9.3.2 Dynamic
Equation in Modal Coordinates

- 9.3.2.1 Modal Coordinates

- 9.3.2.2 Motion Equation in Modal Coordinates

- 9.3.3 Modal Analysis with Damping Neglected

- 9.3.4 Modal Analysis with Damping Included

- 9.4    Evaluation of Structure Damping

- 9.4.1 Modal Damping

- 9.4.2 Damping Matrix

- 9.5    Finite Element Nonlinear Analysis

 

Chapter 10  Damping of Sandwich Materials and Structures

- 10.1    Modelling the Dampig of Sandwich Composite Materials and

             Structures

- 10.1.1 Stress Field in Sandwich Composite Materials

- 10.1.2 In-Plane Strain Energy

- 10.1.3 Transverse Shear Strain Energy

- 10.1.2 Damping of a Sandwich Composite Structure

- 10.2     Experimental Investigation of the Damping of Sandwich





              Materials

- 10.2.1 Materials

- 10.2.2  Determination of the Constitutive Damping Parameters

- 10.3      Results and Discussion

- 10.3.1 Determination of the Dynamic Characteristics of the Foams

- 10.3.1.1 Test Specimens

- 10.3.1.2 Energies Stored in the Test Specimens and Procedure

- 10.3.1.3 Dynamic Characteristics of the Foams

- 10.3.2 Analysis of the Damping of Sandwich Materials

- 10.3.2.1 Introduction

- 10.3.2.2 Mode Shapes

- 10.3.2.3 Damping of the Sandwich Materials       

- 10.4        Characteristic Factors of the Damping of Sandwich Materials

- 10.4.1 Influence of the Shear Modulus of the Foam Core

- 10.4.2 Energies Dissipated in the Core and Skins

- 10.4.3 Effect of the Core Thickness on the Damping of Sandwich
             Materials

- 10.5        Conclusions

 

Chapter 11    General Formulation of Damping of Composite Materials 

                      and Structures

- 11.1     Materials

- 11.2     Modelling Damping of Laminate Beams and Rectangular
               Plates

- 11.3      Damping Modelling using Finite Element Analysis

- 11.3.1 Introduction

- 11.3.2 Stress Field in Composite Materials

- 11.3.3  In-Plane Strain Energy

- 11.3.4  Transverse Shear Strain Energy

- 11.3.5 Damping of a Composite Structure

- 11.3.6 Procedure for Evaluating the Damping of Composite Structure

- 11.4      Investigation of the Damping of Composite Materials

- 11.4.1 Determination of the Constitutive Damping Parameters

- 11.4.2 Damping of the Glass Fibre Laminates

- 11.4.3 Damping Comparison between Taffeta Laminates, Serge 

             Laminates and Cross-Ply Laminates

- 11.4.4 Damping of the Unidirectional Glass Fibre Laminates



- 11.5        Dynamic Response of a Composite Structure

- 11.6  Conclusions

 

References